Structure of stability sets and asymptotic stability sets of families of linear differential systems with parameter multiplying the derivative: I

We consider families of linear differential systems continuously depending on a real parameter. The stability (respectively, asymptotic stability) set of such a family is defined as the set of all values of the parameter for which the corresponding systems in the family are stable (respectively, asymptotically stable). We show that a set on the real axis is the stability (respectively, asymptotic stability) set of some family of this kind if and only if it is an F _σ -set (respectively, an F _σδ -set). For families in which the parameter occurs only as a factor multiplying the matrix of the system, their stability sets are exactly F _σ -sets containing zero on the real line. The asymptotic stability sets of such families will be described in the second part of the present paper.